Math  /  Trigonometry

Question1 HW HW Score: 96.05%,36.596.05 \%, 36.5 of 38 Question 38, *5.5.85 points Points: 0 of 1 Save
Use a calculator to solve the equation on the interval [0,2π)[0,2 \pi). sinx=0.8634\sin x=-0.8634
What are the solutions in the interval [0,2π)[0,2 \pi) ? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. x=x= \square (Type your answer in radians. Round to four decimal places as needed. Use a comma to separate answers as needed.) B. There is no solution.

Studdy Solution

STEP 1

1. We are solving the trigonometric equation sinx=0.8634\sin x = -0.8634.
2. The solution is required in the interval [0,2π)[0, 2\pi).
3. We will use a calculator to find the inverse sine and determine the solutions.

STEP 2

1. Find the reference angle using the inverse sine function.
2. Determine the solutions in the specified interval.
3. Verify the solutions are within the interval [0,2π)[0, 2\pi).

STEP 3

Use a calculator to find the reference angle by taking the inverse sine of the positive value of the given sine:
θ=sin1(0.8634)\theta = \sin^{-1}(0.8634)
Calculate θ\theta using a calculator, ensuring the calculator is set to radian mode.

STEP 4

Since sinx=0.8634\sin x = -0.8634, we know that xx is in the third or fourth quadrant, where sine is negative.
1. For the third quadrant, the angle can be found using: $ x = \pi + \theta \]
2. For the fourth quadrant, the angle can be found using: $ x = 2\pi - \theta \]
Calculate both values of xx using the reference angle θ\theta obtained in Step 1.

STEP 5

Ensure the calculated values of xx are within the interval [0,2π)[0, 2\pi). If any value is outside this range, adjust accordingly.

STEP 6

Verify the solutions by substituting back into the original equation to ensure they satisfy sinx=0.8634\sin x = -0.8634.
The solutions for xx in the interval [0,2π)[0, 2\pi) are:
x=3.9981,5.4081 x = \boxed{3.9981, 5.4081}

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